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					                                                      International Journal of Advances in Science and Technology,
                                                                                                 Vol. 4, No 4, 2012



       Cototal Domination in Semitotal- Point Graph
                                B.Basavanagoud*, Prashant. V. Patil and Sunilkumar. M. Hosamani

                            Department of Mathematics, Karnatak University, Dharwad-580 003,Karnataka, India

                                              Email*: b.basavanagoud@gmail.com




                                                       Abstract

     Let               be a graph. Then the semitotal-point graph is denoted by                 . Let the
     vertices and edges of be called the elements of . A dominating set         of a graph   is a cototal
     dominating set of if                 has no isolated vertices. The cototal domination number of
     denoted by          of is the minimum cardinality of a cototal dominating set of . In this paper
     we study the cototal domination in semitotal-point graphs and obtain many bounds of               in
     terms of elements of     but not the elements of . Also its relationship with other domination
     parameters are established.

     2000 Mathematics Subject Classification: 05C69.

     Keywords: domination number, cototal domination number.

     1 Introduction.

                Throughout this paper by the word graph we mean a finite, nontrivial, undirected, connected,
     without loops or multiple edges. Let the vertices and edges of a graph       be called the elements of .
     For notations and terminology we follow Harary [4].
               The neighborhood of a vertex in is the set               consisting of all vertices which are
     adjacent with . A set              is a dominating set, if every vertex in             is adjacent to some
     vertex in . The domination number              of , is the minimum cardinality of a dominating set. A
     dominating set of a graph is a cototal dominating set of if every vertex                           is not an
     isolated vertex in the induced subgraph            . The cototal domination number              of , is the
     minimum cardinality of a cototal dominating set [7].
               For any graph                , semitotal-point graph                 is the graph whose vertex
     set is the union of vertices and edges in which two vertices are adjacent if and only if they are adjacent
     vertices of or one is a vertex and other is an edge of incident with it [8].
               We now define the cototal domination number of the semitotal-point graph as follows:
               A dominating set      of a graph    is a cototal dominating set of      if                 has no
     isolated vertices. The cototal domination number of denoted by                       of    is the minimum
     cardinality of a cototal dominating set of .

                The following figure shows the formation of     and    .




Special Issue                                       Page 109 of 114                                      ISSN 2229 5216
                                                                                International Journal of Advances in Science and Technology,
                                                                                                                           Vol. 4, No 4, 2012

                                                                                                                              1
                                            1

                                            a                                                                                              a

                G:                                                                           T2(G):
                                                                                              1                  d
                                        2                                                                                         2                 b
                               d                    b


                          4                                     3                                               4
                                        c                                                                                                   3

                                                                                                                                  c
                                                                                    Figure 1

                 In ,               , therefore                                                .
                In ,                 ,          ,                                   and so on. Therefore                                       .

                2 Main Results
                 In the following theorem, we state the exact values of cototal domination number of                                           of
                some standard class of graphs.
                Theorem 2.1
                        For any path                ;                       ,                        .
                        For any cycle                   ;                       ,                        .
                         For any complete bipartite graph                                            ;               ,                 .
                         For any complete graph                                     ;       ,



                  For any wheel                 ;                       ,                       .

                The following upper bound of                                        is immediate from the above Theorem 2.1.

                Theorem 2.2 For any graph                   ,                                       . Further the equality holds for
                for           and   is odd.

                Now, we establish another upper bound for                                            when            is a tree.

                Theorem 2.3 For any tree                            ,

                Proof. Let     be a dominating set of and           be a cutvertex in . Let
                                     be the corresponding edge vertices of in . Then, every
                                       may or may not be an isolated vertex in                                 . Now, it follows that,         ;




Special Issue                                                           Page 110 of 114                                                    ISSN 2229 5216
                                                             International Journal of Advances in Science and Technology,
                                                                                                        Vol. 4, No 4, 2012


                             form a cototal dominating set of            . Hence                          .
                 The next theorem gives an upper bound of                     in terms of order and minimum degree
                of .

                Theorem 2.4 If                                , then                          , where              is the
                minimum degree of          .

                Proof. Let   be a vertex of minimum degree                  such that                              . Let
                                    be the set of edge vertices in          with respective the edges


                of   . Suppose                  . Then each edge vertex forms a cycle         in      and by definition
                of   , has               vertices and     edges. Therefore,
                                                                       . Hence                                 .

                In a graph         if                    , then       is called a pendant vertex of .

                Theorem 2.5 For any           tree                ,                     , where        is the number of
                vertices adjacent to pendant vertices.

                Proof. Suppose is a tree with vertices and                                         be the set of vertices
                adjacent to pendant vertices of with size                     . Let                        be the set of
                edge vertices in        with respective to the edges                      of . Since each edge
                vertex     ;            is adjacent to exactly two vertices and forms a cycle of length three
                in . Then,                  contains at least one isolated vertex. Let    be the set of such
                vertices. Since                          and also             and         is the set of all
                pendant
                vertices of , therefore          forms a cototal dominating set of . Hence
                                           number of pendant vertices of                .

                Theorem A [3]. If         is a graph without isolated vertices of order , then                      .
                Theorem 2.6 If          contains a pendant vertex, then                  .

                Proof. Let   be a vertex of degree one and the set                             be the set of edge
                vertices in with respective the edges               of . Now we consider the
                following cases:
                Case 1. If           and is a tree then by Theorem 2.3,           .

                Case 2. If                   and is not a tree. Let        and starting from , consider
                the alternate vertices of and that form a dominating set of . Since each vertex of
                belongs to at least one edge vertex in , therefore is a cototal dominating set of .
                Since is also a dominating set of , therefore by Theorem A,
                                                     .
                Theorem 2.7 If          is a tree in which the number of pendant vertices is same as the
                number of cutvertices and every cutvertex is adjacent to a pendant vertex, then
                             , where is the number of pendant vertices or number of cutvertices of .




Special Issue                                             Page 111 of 114                                               ISSN 2229 5216
                                                          International Journal of Advances in Science and Technology,
                                                                                                     Vol. 4, No 4, 2012


                 Proof. Let be a tree with the above stated property. Then                  . Let     be
                the cototal dominating set of . If set                are the edges of , then the set
                                     will be corresponding edge vertices of        . Since each    ;                    is
                    , then contains a cycle of length three and every               ;             is adjacent
                the set of all such vertices. Then                has no isolated vertex. On the other hand,
                let           , then the set                  forms a cototal dominating set of and
                                       , which is not possible. So              . Now, it follows that every
                has at least one neighbor in and                                                                    .
                Thus the result.

                Theorem B [5]. For any connected graph            of order ,
                where         is the maximum degree of        .

                Next we obtain a lower bound of               in terms of order, size and maximum degree of
                 .

                Theorem 2.8 For any             graph     ,


                Proof. Since every cototal dominating set is a dominating set, therefore                                .
                Also by Theorem B, we get the required result.

                Theorem C [7]. For any graph       ,                  if and only if each component of       is a
                star.

                Theorem 2.9 In a tree            , if each component is a star, then                         .

                Proof. Let the set                      be the set of all center vertices of stars in a tree. Then,
                                                         . But by Theorem C,                   . Hence from these
                two we get the result.

                Theorem 2.10 If             is an independent dominating set of both        and    , then
                              .

                Proof. Let                          be the set of independent vertices of which also a
                dominating set of and        . Also by definition of , each edge vertex forms a    and
                is independent, therefore                 has no isolated vertex. Therefore                         .

                 A dominating set of a graph , is called a maximal dominating set, if       is not a
                dominating set of . The maximal domination number          of , is the minimum
                cardinality of a maximal dominating set [6 ].

                Theorem D [6]. For any graph       , exactly one of these holds:




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                                                              International Journal of Advances in Science and Technology,
                                                                                                         Vol. 4, No 4, 2012


                Theorem 2.11 For any graph         ,                                .

                Proof. Let be a maximal dominating set of and if for a vertex                         is an
                isolated vertex, then          is a cototal dominating set of . If                   is not
                an isolated vertex, then itself forms a cototal dominating set of . Hence it follows
                from result      of Theorem D,                                        . Thus the result.

                Theorem 2.12 For any graph             with            ,                      .

                Proof. Let be a connected graph. Let      and     be the cototal dominating sets of
                and respectively. Suppose                                           forms a dominating set
                of . Let set                 be the set of edge vertices in with respect to the edges
                               of   . Then every         ;                 is adjacent with exactly two point vertices
                of in . Since                    and               has no isolated vertices, therefore it
                follows that forms a cototal dominating set of . Hence
                                                   . Therefore                    .

                 A dominating set is a total dominating set if the induced subgraph      has no isolated
                vertex. The total domination number         of a graph is the minimum cardinality of a
                total dominating set. The vertex independence number         is the maximum cardinality
                among the independent set of vertices of .

                Theorem 2.13 For any graph                     ,                                  .

                Proof. Let be any graph without isolates and be a total dominating set of . Suppose
                the induced subgraph              has no isolates then is a cototal dominating set of .
                If there exists a set        such that each vertex in is an isolated vertex in
                                . Thus   forms a cototal dominating set of and . Thus by above
                Theorem 2.12,                              .

                Nordhaus-Gaddum type results.

                Theorem 2.14 Let a graph                     and its complement         be connected. Then


                                                                   .



                                                         Acknowledgement
                   This research was supported by UGC-SAP DRS-II New Delhi, India: for 2010-2015.

                References

                [1] B.Basavanagoud, S.M.Hosamani and S.H.Malghan, Domination in semitotal-point
                graphs, J.Comp. and Math.Sci., Vol.1(5)(2010), 598-605.




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                                                     International Journal of Advances in Science and Technology,
                                                                                                Vol. 4, No 4, 2012

                [2] E. J. Cockayne, R.M.Dawes and S. T. Hedetniemi, Total domination in graphs,
                Networks , 10(1980), 211-219.

                [3] E. J. Cockayne and S. T. Hedetniemi, Towards a theory of domination in Graphs,
                Networks , 7(1977), 247-261.

                [4] F. Harary, Graph Theory, Addison-Wesley, Reading, Mass, (1969).

                [5] T.W.Haynes, S.T.Hedetniemi and P.J.Slater, Fundamentals of domination in graphs,
                Marcel Dekker, Inc, New York (1998).

                [6] V.R.Kulli and B.Janakiram, The maximal domination number of a graph, Graph Theory
                Notes of New York, New York Academy of Sciences, XXXIII(1997), 11-13.

                [7] V.R.Kulli, B.Janakiram and R.R.Iyer, The cototal domination number of a graph
                J.Discrete Mathematical Sciences and Cryptography, 2(1999), 179-184.

                [8] E.Sampathkumar and S.B.Chikkodimath, The semi-total graphs of a graph-I, Journal
                of the Karnatak University-Science, XVIII(1973), 274-280.


     Authors Profile

                                        Dr. B.Basavanagoud received his both MSc and Ph.D degree in
                              Mathematics from Gulbarga University, Gulbarga, India. Currently he is
                              working as Professor and Chairman, Department of Mathematics, Karnatak
                              University, Dharwad. He has published many research papers in reputed
                              national and international jourmnals. His area of interest lies mainly in Graph
                              Theory. He is also life member of The Indian Mathematical Society, The
                              Indian Science Congress Association and International Journal of
                              Mathematical Sciences and Engineering Applications. He chaired the session
                              in the International Congress of Mathematicians(ICM-2010) and delivered a
                            contributed talk at the same




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